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Random Field Theory Consider a statistic image as a lattice representation of a continuous random fieldConsider a statistic image as a lattice representation of a continuous random field Use results from continuous random field theoryUse results from continuous random field theory Consider a statistic image as a lattice representation of a continuous random fieldConsider a statistic image as a lattice representation of a continuous random field Use results from continuous random field theoryUse results from continuous random field theory Lattice representation

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Functional Imaging Data The Random Fields are the component fields,The Random Fields are the component fields, Y = Xw +E, e=E/σ Y = Xw +E, e=E/σ We can only estimate the component fields, usingWe can only estimate the component fields, using estimates of w and σ estimates of w and σ To apply RFT we need the RESEL count which requires smoothness estimatesTo apply RFT we need the RESEL count which requires smoothness estimates The Random Fields are the component fields,The Random Fields are the component fields, Y = Xw +E, e=E/σ Y = Xw +E, e=E/σ We can only estimate the component fields, usingWe can only estimate the component fields, using estimates of w and σ estimates of w and σ To apply RFT we need the RESEL count which requires smoothness estimatesTo apply RFT we need the RESEL count which requires smoothness estimates

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Cluster and Set-level Inference We can increase sensitivity by trading off anatomical specificityWe can increase sensitivity by trading off anatomical specificity Given a voxel level threshold u, we can computeGiven a voxel level threshold u, we can compute the likelihood (under the null hypothesis) of getting n or more connectedcomponents in the excursion set ie. a cluster containing at least n voxels the likelihood (under the null hypothesis) of getting n or more connectedcomponents in the excursion set ie. a cluster containing at least n voxels CLUSTER-LEVEL INFERENCE CLUSTER-LEVEL INFERENCE Similarly, we can compute the likelihood of getting cSimilarly, we can compute the likelihood of getting c clusters each having at least n voxels clusters each having at least n voxels SET-LEVEL INFERENCE SET-LEVEL INFERENCE We can increase sensitivity by trading off anatomical specificityWe can increase sensitivity by trading off anatomical specificity Given a voxel level threshold u, we can computeGiven a voxel level threshold u, we can compute the likelihood (under the null hypothesis) of getting n or more connectedcomponents in the excursion set ie. a cluster containing at least n voxels the likelihood (under the null hypothesis) of getting n or more connectedcomponents in the excursion set ie. a cluster containing at least n voxels CLUSTER-LEVEL INFERENCE CLUSTER-LEVEL INFERENCE Similarly, we can compute the likelihood of getting cSimilarly, we can compute the likelihood of getting c clusters each having at least n voxels clusters each having at least n voxels SET-LEVEL INFERENCE SET-LEVEL INFERENCE

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SummarySummary We should correct for multiple comparisonsWe should correct for multiple comparisons We can use Random Field Theory (RFT)We can use Random Field Theory (RFT) RFT requires (i) a good lattice approximation to underlying multivariate Gaussian fields, (ii) that these fields are continuous with a twice differentiable correlation functionRFT requires (i) a good lattice approximation to underlying multivariate Gaussian fields, (ii) that these fields are continuous with a twice differentiable correlation function To a first approximation, RFT is a Bonferroni correction using RESELS.To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct for the volume of interest.We only need to correct for the volume of interest. Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference. We should correct for multiple comparisonsWe should correct for multiple comparisons We can use Random Field Theory (RFT)We can use Random Field Theory (RFT) RFT requires (i) a good lattice approximation to underlying multivariate Gaussian fields, (ii) that these fields are continuous with a twice differentiable correlation functionRFT requires (i) a good lattice approximation to underlying multivariate Gaussian fields, (ii) that these fields are continuous with a twice differentiable correlation function To a first approximation, RFT is a Bonferroni correction using RESELS.To a first approximation, RFT is a Bonferroni correction using RESELS. We only need to correct for the volume of interest.We only need to correct for the volume of interest. Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.Depending on nature of signal we can trade-off anatomical specificity for signal sensitivity with the use of cluster-level inference.